The common ancestor type distribution of a $\Lambda$-Wright-Fisher process with selection and mutation

TL;DR

This paper develops a graphical representation for the ancestor type distribution in a $ ext{Lambda}$-Wright-Fisher process with mutation and selection, extending previous models to include heavy-tailed offspring distributions.

Contribution

It introduces a novel graphical method based on the ancestral selection graph for $ ext{Lambda}$-Wright-Fisher processes, generalizing prior results to heavy-tailed offspring distributions.

Findings

Representation of ancestor type distribution using equilibrium tail probabilities

Identification of a pathwise Siegmund dual process

Characterization of tail probabilities via hitting probabilities

Abstract

Using graphical methods based on a `lookdown' and pruned version of the {\em ancestral selection graph}, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from Lenz, Kluth, Baake, and Wakolbinger (Theor. Pop. Biol., 103 (2015), 27-37) to the case of heavy-tailed offspring, directed by a reproduction measure $\Lambda$. The representation is in terms of the equilibrium tail probabilities of the line-counting process $L$ of the graph. We identify a strong pathwise Siegmund dual of $L$, and characterise the equilibrium tail probabilities of $L$ in terms of hitting probabilities of the dual process.